Fixed-core construction avoiding midpoint intersections ($|\mathcal{A}|=2\left(\genfrac{}{}{0ex}{}{3p-1}{p-1}\right)$)

Fix a $(p+1)$-set $B\subseteq [4p]$. Take every $2p$-set $B\cup X$ with $X$ a $(p-1)$-subset of the remaining $3p-1$ points, together with the complements of all these sets. Within either half, intersections have size at least $p+1$. Across the halves they have size $|X\setminus Y|\le p-1$. Thus no intersection has size $p$, and the two disjoint halves give the displayed family size.